www.scielo.br/rbg
ADAPTIVE SIGNAL DECOMPOSITION AND FILTERING USING BINOMIAL FILTERS
APPLIED TO THE GROUND-ROLL ATTENUATION
Cristian D. Ariza A.
1and Milton J. Porsani
2ABSTRACT.The ground-roll is a type of noise normally present in land seismic data. It strongly harms the signal-to-noise ratio, and interferes in several stages of the seismic data processing, strongly affecting the final quality of the obtained seismic images. For the attenuation of the ground-roll we propose an adaptive filtering method that uses binomials filters obtained from the convolution of pairs of dipoles(1, c) and (c, −1). The coefficient c corresponds to the first coefficient of the prediction error operator, obtained through the Burg algorithm. This filtering method allows for the decomposition of signals in frequency bands, from the lowest frequency band to the highest band. The method is applied in adaptive form, in sliding windows of time, adapting themselves to the variations in the frequency content of the data. Its implementation and use in seismic data processing is relatively simple and computationally efficient.
Keywords: seismic noise, signal-to-noise ratio, adaptive filters, Burg algorithm, seismic signal decomposition.
RESUMO.Oground-roll´e um tipo de ru´ıdo normalmente presente nos dados s´ısmicos terrestres. Ele prejudica muito a raz˜ao sinal-ru´ıdo e interfere em v´arios est´agios do processamento de dados s´ısmicos, afetando fortemente a qualidade final das imagens s´ısmicas obtidas. Para atenuac¸˜ao desse ru´ıdo, propomos um m´etodo de filtragem adaptativa que utiliza filtros binomiais constru´ıdos a partir da convoluc¸˜ao de pares de dipolos(1, c) e (c, −1). O coeficiente c corresponde ao primeiro coeficiente do operador de erro de predic¸˜ao e ´e obtido atrav´es do algoritmo de Burg. Este m´etodo de filtragem adaptativa permite a decomposic¸˜ao de sinais em bandas de frequˆencia, desde a banda de mais baixa frequˆencia at´e a de mais alta. O m´etodo ´e aplicado na forma adaptativa, utilizando janelas deslizantes de tempo, adaptando-se `as variac¸˜oes do conte´udo de frequˆencia dos dados ao longo do registro s´ısmico. Sua implementac¸˜ao e uso, no processamento de dados s´ısmicos, ´e relativamente simples e computacionalmente eficiente.
Palavras-chave: ru´ıdos s´ısmicos, relac¸˜ao sinal-ru´ıdo, filtragem adaptativa, algoritmo de Burg, decomposic¸˜ao do sinal s´ısmico.
1Universidade Federal da Bahia (UFBA), Center for Research in Geophysics and Geology (CPGG), Rua Bar˜ao de Geremoabo s/n, 40170-115 Salvador, BA, Brazil. Phone: +55(71) 3283-8530 – E-mails: cristian.ariza@ufba.br; arizacristian@gmail.com
2Universidade Federal da Bahia (UFBA), Center for Research in Geophysics and Geology (CPGG) and National Institute of Science and Technology of Petroleum Geophysics (INCT-GP/CNPq), Rua Bar˜ao de Geremoabo s/n, 40170-115 Salvador, BA, Brazil. Phone: +55(71) 3283-8530; Fax: +55(71) 3283-8501 – E-mail: porsani@ufba.br
INTRODUCTION
One of the most significant challenges in processing of seismic data is to filter different types of noises. Ground-roll is one of the main types of coherent noise in land seismic data. It has the sig-nificant characteristics of relatively low velocity, low frequency, high amplitude and strong energy (Sheriff, 2002). Because of its dispersive nature and low velocity, ground-roll masks the shal-low reflections at near offsets and deep reflections at far offsets (Saatcilar & Canitez, 1988; McMechan & Sun, 1991; Saatcilar & Canitez, 1994; Henley, 2003) and also distorts reflection events by interfering with them.
Ground-roll is one of the main coherent noises in petroleum seismic exploration, many methods have been introduced to at-tenuate this type of noise. Although the right choice of attenu-ation techniques is a matter of trial and error (Sheriff & Geldart, 1995). The conventional methods can be divided into two groups. The first one can be summarized to filter method which is based on suppression of undesired parts of recorded data in the spec-tral domain, including high-pass and band-pass filtering,f − k filtering (Embree et al., 1963; Treitel et al., 1967; Yilmaz, 2001) and the adaptive ground-roll attenuation method (Wang et al., 2012; Hosseini et al., 2015). These methods have their limita-tions. High-pass and band-pass filter may eliminate the low fre-quency component of effective waves since the frefre-quency bands of ground-roll noise and reflections are often overlapped (Sirgue, 2006). The conventional f − k filter would cause serious dis-tortion of effective waves when the energy of ground-roll noise is much stronger than that of reflections (McMechan & Sun, 1991; Liu, 1999; Tokeshi et al., 2006). The other one is wave field sep-aration method based on ground-roll noise extraction and arith-metical subtraction of it from the raw shot gather in the t − x domain, including Wiener-Levinson algorithm (Karsli & Bayrak, 2004), Karhunen-Lo`eve (K-L) transform (G´omez Londo˜no et al., 2005), wavelet transform (Deighan & Watts, 1997) and Radon transform (Russell et al., 1990a,b).
There are different methods of decomposing a seismic signal used for suppressing the ground-roll: Empirical Mode Decom-position (EMD), developed by Huang et al. (1998) and used by Ferreira et al. (2013); singular value decomposition, SVD used by Porsani et al. (2010), Wavelet decomposition used by Deighan & Watts (1997), decomposition by filtering frequency bands with binomial operators justified by Akansu & Haddad (2001) and Vetterli & Herley (1992) used by Ariza & Porsani (2015) for at-tenuation ground-roll.
Filtering with binomial operators (Haddad, 1971; Vetterli & Herley, 1992; Akansu & Haddad, 2001), enable decomposition
and perfect reconstruction of the signal through the linear combi-nation of its components. The decomposition of a signal is made into frequency bands by means of a matrix operators X, which is obtained by weighting each column of the matrixXjn(obtained through the dipoles(1, 1) and (1, −1)). This procedure allows for the original signal shifted to the position Sn = Sn∗δn−a. To recover the signal in the initial position just delay it fora samples. It can be shown that the construction of the matrixX, through the dipoles(1, 1) and (1, −1), is just a special case of a general breakdown binomial with dipoles(α, β) and (β, −α) where α andβ can be real or complex (in this work we will only consider the case real), where the values ofα and β beta can be arbitrarily chosen or may be obtained by the features given itself, creating infinite possibilities of decomposition. In the present paper we compute the dipoles adaptively, with(1, c) and (c, −1), where c is the coefficient of the prediction error operator obtained by using the Burg algorithm.
THEORY
Following is presented a form of general binomial decomposi-tion with dipoles(α, β) and (β, −α), where α and β can be real or complex (in this text we will only consider the real case). The dipoles(1, 1) and (1, −1) are a special case of this gen-eral representation (Boyd et al., 2001; Severo, 2008; Severo & Schillo, 2009).
In the case of a binary decomposition of order 1 (N = 1), the operators matrixX is written as follows:
X = α β β −α = yT 0 yT 1 =x0 x1 (1) You can verify thatX2 = λI. For the case where N = 1, λ = α2+ β2, so the matrixX−1is:
X−1= 1 α2+ β2 α β β −α (2)
The operator matrix X is obtained, for example, selecting the first column ofX−1(a = 0). This results in
X0= ⎡ ⎢ ⎢ ⎢ ⎣ α2 α2+ β2 β2 α2+ β2 αβ α2+ β2 −αβ α2+ β2 ⎤ ⎥ ⎥ ⎥ ⎦ (3)
It should be noted that if the columns of the matrix X are stacked, the result is(1, 0)T.
Applying binary decomposition of order 1(N = 1) to a discrete signal {Sn} = {s0, . . . , sM−1} is equivalent to performing the convolution of the original signal with each of the columns of the operator matrix X0. The signal recovery is ob-tained by a simple stacking of the signal components.
{Rn} = {Sn} ∗ { X0 0} + {Sn} ∗ { X10} = {Sn} ∗ { X0 0+ X10} = {Sn} ∗ (1, 0)T = {S n} ∗ {δn} = {Sn} (4)
The operators matrixX for any order N (where X have N + 1 columns) is calculated as follows:
Xr
n= (α, β)∗(N−r)∗ (β, −α)∗r (5) for0 ≤ r ≤ N. Remember that
x∗n= x ∗ x ∗ x ∗ · · · ∗ x ∗ x n
(6) denotes n-times convolution, where x∗0 = δ0, x∗1 = x andδ0is the Kronecker delta.
The matrix generated, as shown previously, keeps the prop-erty of orthogonality between rows and columns, i.e.,yTi xj = 0 for i = j and yTixj = λN for i = j. That is
X2 = λNI (7)
where
λ = α2+ β2 (8)
calculating the inverse matrix as X−1= 1
λNX (9)
The propertyX2 = λNI is particularly important in sig-nal decomposition and filtering. We note that, despite of a scale factor,λN,X matrix is its own inverse. This property provides N + 1 possibilities to decompose the original signal, each re-lated to a particular column of the matrix Xa(a = 0, 1, . . ., N). The signal can be decomposed or restored as a linear combi-nation of components, obtained by convolution of the original signal with the columns of the matrix Xa. Each component, { Sr
n} = { Xnr} ∗ {Sn}, will have a different frequency con-tent depending on the valuesα and β selected.
The values of α and β can be arbitrary or can be calcu-lated adaptively, using local characteristics of the data{Sn} = {s0, . . . , sM−1}. Consequently, the method is versatile and relatively simple to implement. It may be used for signal analysis or filtering, and there are many ways to perform the decomposi-tion of a signal.
RESULT
In order to get local information from the data, the recursive Burg algorithm was used to obtain the first coefficient of the prediction error for a specified window’s length (Burg, 1967). It is possible to generate dipoles(1, c) and (, −1), where c is the first pre-diction error coefficient, for a given window. Taking into account the minimum phase property of the linear unit error prediction filters when they are calculated using the least squares method (Appendix A of Chu, 2004), the first coefficient is always negative less than one, then the first dipole(1, c) is a differentiation oper-ator allowing the capture of high frequency information (depend-ing on the window width) and the second dipole(c, −1) (both negative) would be an operator of integration which allows cap-turing low frequency information.
We use a 96 channel common-shot-point gather (Fig. 6A) acquired in the North-west Brazil (Tacutu Basin) to demonstrate the feasibility and applicability of the proposed method. The geo-phone interval is 50 m. The length of the record is 4000 ms with a 4 ms sample interval.
There were tested different window sizes (from 20 ms to 2000 ms), achieving the best contrast between areas with and without ground-roll with a 200 ms window. Figure 1 shown the values of the coefficient c, for the shot gather shown in Figure 6A. The presence of the ground-roll, corresponding to the low frequency areas, is marked by the lower amplitude values of the c coefficients. More information about how was construct this graph, see appendix.
Figure 1 – Map of the prediction error coefficients, used to generate the
The algorithms used to trace decomposition are shown below (Algorithm 1 and Algorithm 2).
Signal decomposition and reconstruction
To test the decomposition and perfect reconstruction of the signal it was used a level of decompositionN = 7 weighted by the first column of the inverse matrix (a = 0). So the seismic traces in Figure 6A is decomposed into 8 panels with different frequency content, from highest to lowest frequency. Figure 2 shows the corresponding average amplitude spectra. We note that gener-ally the decomposed panels have lower amplitude spectra than the original.
Figure 3 shows four ( S1, S3, S5 and S7) of the eight decomposed panels. It can be seen that each decomposed shot gather has different frequency content. Air wave (or sound wave) noise, with an apparent velocity of about 340 m/s, can be iden-tified in S1 (Fig. 3A). It is clear that air wave is a type of
non-dispersive coherent noise, as same as body wave. As seen in panel S7, (Fig. 3D), reflections are heavily contaminated by largely dispersed ground-roll noise. The differences between the firsts panels, corresponding to higher frequencies, are minimal. The stack of all decomposed panels reconstruct the original shot gather. The error of the reconstruction is due to the numerical computational artifacts (Fig. 4).
Ground-roll attenuation
The weak reflections are invisible due to the interference of the strong ground-roll noise (Fig. 6A). Furthermore, the reflected events in the shot gather are even truncated due to the high-amplitude of the ground-roll. The blue line in Figure 2 shows the average amplitude spectra of the ground-roll. We can see that mostly of its energy is concentrate between frequencies 1 Hz to 10 Hz, overlapping with the frequencies content of the reflections, with occur dominantly between 10 Hz to 50 Hz. Figure 3D shown
Algorithm 1 – Trace decomposition.
Require: S(ns): Input trace (number of samples); Lw: long window; N: level of decomposition; 1: initialDS(ns, N + 1) = 0.0; x(Lw) = 0.0; Y(Lw, N + 1) = 0.0; z(ns) = 0.0 2: fori = 1, ns − Lw + 1 do
3: x ← S(i : i + Lw − 1);
4: Y ← Compute window decomposition of x ;
5: DS(i : i + Lw − 1, :) ← DS(i : i + Lw − 1, :) + Y; 6: z(i : i + Lw − 1) ← z(i : i + Lw − 1) + 1;
7: end for
8: DS ← DS(j, :) ← DS(j, :)/z(j); j = 1, . . ., ns 9: return Decomposed traceDS(ns, N + 1)
Algorithm 2 – Window decomposition.
Require: x(Lw): Input (number of samples); N: level of decomposition; a = 0: Weighting column. 1: initialX(N + 1, N + 1) = 0; X(N + 1, N + 1) = 0; Y(Lw + N, N + 1) = 0; 2: if|x| > 0 then
3: Computec with Burg algorithm
4: X ← Xnr= (1, c)∗(N−r)∗ (c, −1)∗r r = 0, 1, . . ., N 5: λ ← α2+ β2;α = 1 ; β = c 6: X−1← X/λN 7: X ← X(:, j) ← X(:, j) · X−1(j, a); j = 0, 1, . . ., N 8: Y ← Y (:, j) ← x ∗ X(:, j); j = 0, 1, . . ., N 9: else 10: Y = 0.0 11: end if
Figure 2 – Average amplitude spectra of the shot gather decomposition by using dipoles(1, c) and (c, −1) for N = 7.
that the ground-roll may be successfully separated out from the original shot gather. Observing the amplitude spectra of the orig-inal and decomposed shot gathers (Fig. 2), we note that the dom-inant low frequency content may be formed by adding panels
S6 and S7. Hence, a good option to mitigate the ground-roll is to reconstruct the signal by adding panels from S0 to S5, which we nominate as DB1c result. That is equivalent to subtract
S6and S7from the original shot gather.
To validate the DB1c result we applied two others commonly used methods to filter the shot gather data: trapezoidal frequency band-pass filter (f1 = 10 Hz, f2 = 20 Hz, f3 = 60 Hz, andf4 = 70 Hz) and f − k filtering. Figure 5 show the av-erage amplitude spectra of the filtered shot gathers. Since the frequencies bands of ground-roll and reflections are overlapped, the band-pass filter attenuate both, the low frequency component of the ground-roll and the signal. Thef − k filtering still retains strong ground-roll noise in the frequency band from 0 Hz to 20 Hz, whereas the proposed method (DB1c) removed most part of the ground-roll.
Figure 6 shows comparisons between the original shot gather and results obtained by using frequency domain methods. All three methods provide ways to attenuate ground-roll, but band-pass filter has significant decrease amplitude drawback. f − k filtering causes serious distortion of the reflection and the adap-tive binomial filtering is more effecadap-tive to the ground-roll attenu-ation, preserving better the reflections.
CONCLUSIONS
We presented an adaptive binomial filtering method where the fil-ters are obtained by convolving dipoles. These dipole coefficients
are associated with the prediction error operator and are computed by using the Burg algorithm. We show that the matrix associated with the binomial filter may be used to perform signal decompo-sition and/or filtering the data. We applied the method to filter the ground-roll in a shot gather. The obtained results showed the new approach is more effective when compared with the commonly used filtering methods working in the frequency domain. Below we summarize the main general characteristics of the proposed adaptive binomial filter:
• The implementation of the bank of binomial filters, at any level, allows for the decomposition and also for a perfect reconstruction of seismograms;
• Any additional processing may be performed in any of the decomposed signal or panel, which makes the method very versatile;
• The generation of adaptive binomial operator (filter) only involves dipoles convolution and their use in filtering of seismic data is considerably simple and computationally efficient;
• The signal decomposition/filtering is performed trace by trace. That property may be explored by using paralleliza-tion computing for processing of very large data volumes; • The filtering method is robust, easy to implement, com-putationally efficient and requires few parameter setting by the user.
Figure 3 – Four of the eight decomposed shot gathers using the adaptive binomial decomposition method. S1in (A), S3in (B),
Figure 4 – Average amplitude spectra (in decibel scale) of the shot gather and perfect reconstruction results. The red
line, the blue line, and the black line denote spectra of original data, perfect reconstruction and error, respectively. (Red and blue lines are coincident).
Figure 5 – Comparisons of the average amplitude spectra, in linear scale (A) and decibel scale (B), of the original and
filtered shot gathers shown in Figure 6. The red, blue, green and black lines denote spectra of original data and results after filtering using band-pass,f − k, proposed (DB1c) methods, respectively.
Figure 6 – Comparisons between the original shot gather and filtered results. The original shot gather in (A), results after band-pass
Algorithm 3 – Prediction error coefficients map.
Require: S(ns): Entry trace (number of samples); Lw: Window’s length; 1: InitializeMp(ns) = 0.0; x(Lw) = 0.0; y(Lw) = 0.0; z(ns) = 0.0; 2: fori = 1, ns − Lw + 1 do
3: x(1 : Lw) ← S(i : i + Lw − 1); 4: Computec for x(Lw); Burg algorithms 5: y(1 : Lw) ← c;
6: Mp(i : i + Lw − 1) ← Mp(i : i + Lw − 1) + y(Lw); 7: z(i : i + Lw − 1) ← z(i : i + Lw − 1) + 1;
8: end for
9: Mp(j) ← Mp(j)/z(j); j = 1, . . ., ns 10: return map errorMp(ns)
APPENDIX
For the obtaining of the data presented in Figure 1, the following procedure was followed for each seismic trace:
• The coefficient map begins with a value of 0.0;
• The value of the first coefficient error for the window’s length is calculated using the Burg algorithm;
• The coefficient value is added to all the positions of the window’s length in the coefficient map;
• The window is then moved one sample, and the previous two steps are repeated until the end of the trace.
At the end, an average is calculated as a function of the number of values added at each position of the coefficient map. This proce-dure is presented in the Algorithm 3.
ACKNOWLEDGEMENTS
The authors wish to express their gratitude to the Fundac¸˜ao de Amparo `a Pesquisa do Estado da Bahia (FAPESB), INCT-GP, (CNPq), FINEP, PETROBRAS, for financial support.
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Recebido em 13 maio, 2016 / Aceito em 21 maio, 2017 Received on May 13, 2016 / Accepted on March 21, 2017
